arXiv subjects

Ian Calaway

Publications and source records attributed to Ian Calaway.

3 recordsLinked to original sources

arXiv2026

Approval Ballot Triangles and Strict-Sense Ballots

We consider a family of binary triangular arrays, called approval ballot triangles (ABTs), that are in bijection with totally symmetric self-complementary plane partitions (TSSCPPs). These triangles correspond to a ballot process in which voters select their collection of approved candidates rather than voting for a single person. We situate ABTs within the ballot problem literature and then show that a strict-sense ballot can be decomposed into a list of sequentially compatible ABTs.

math.CO↗

arXiv2019

de Finetti Lattices and Magog Triangles

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

math.CO↗

arXiv2018

The Voter Basis and the Admissibility of Tree Characters

When making simultaneous decisions, our preference for the outcomes on one subset can depend on the outcomes on a disjoint subset. In referendum elections, this gives rise to the separability problem, where a voter must predict the outcome of one proposal when casting their vote on another. A set $S \subset [n]$ is separable for preference order $\succeq$ when our ranking of outcomes on $S$ is independent of outcomes on its complement $[n]-S$. The admissibility problem asks which characters $\mathcal{C} \subset \mathcal{P}([n])$ can arise as the collection of separable subsets for some preference order. We introduce a linear algebraic technique to construct preference orders with desired characters. Each vector in our $2^n$-dimensional voter basis induces a simple preference ordering with nice separability properties. Given any collection $\mathcal{C} \subset \mathcal{P}([n])$ whose subset lattice has a tree structure, we use the voter basis to construct a preference order with character $\mathcal{C}$.

math.CO↗